Lecture 8: Hardness of Max-Ek-Independent-Set

Any questions about today's lecture? I will repeat here the hardness reduction:

Given: Max-Label-Cover(K,L) instance $G=(U,V,E)$ with projection constraints ${\pi }_{v\to u}$.

Reduction to a weighted k-uniform hypergraph $H$:
. The vertex set is $V\times \{0,1{\}}^L$ (there are $\mid V\mid 2^{\mid L\mid }$ vertices). We call each $\left\{v\right\}\times \{0,1{\}}^L$ the "block" of vertices corresponding to $v\in V$. We name the strings/sets in this block $A^{\left(v\right)}$.
. The weight on each vertex $A^{\left(v\right)}$ is its $p$-biased weight, $p=1-2/k-\delta$. The total weight in each block is $1$ and the overall total weight is $v$.
. For each pair of vertices $v,vʹ\in V$ sharing a common $u$ (i.e., with $(u,v),(u,vʹ)\in E$), we will put some hyperedges on vertices in the union of the $v$ and $vʹ$ blocks.
. Specifically, we put a hyperedge on the $k$-set $\{A_1^{\left(v\right)},...,A_{k/2}^{\left(v\right)},B_1^{(vʹ)},...B_{k/2}^{(vʹ)}\}$ iff the following condition holds:

${\pi }_{v\to u}(\cap A_i^{\left(v\right)})$ is disjoint from ${\pi }_{vʹ\to u}(\cap B_i^{(vʹ)})$.

(Supertechnical note. The A's should all be distinct, and the B's should all be distinct. Also, we allow $v=vʹ$, in which case the A's and B's should be mutually distinct.)


This construction is polynomial time (since $k$, $\mid K\mid$, $\mid L\mid$ are all constants; supersupertechnical note: we assume $\delta$ is rational).

You should again try to convince yourself of the "completeness" part of the proof: Opt(G) = 1 implies Opt(H) $\ge p=1-2/k-\delta$.